Signals and Systems: Chapter 1 ContinuousTime Signals

• Examples of signals and systems Electrical Systems – Voltage divider • Input signal: x = 5V • Output signal: y = Vout • The system output is a fraction of the input ? = ?2 ?1+?2?– Mult

University of Arkansas

ELEG 3124 SYSTEMS AND SIGNALS

Ch 1 Continuous-Time Signals

Dr Jingxian Wu wuj@uark.edu

(These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.)

EE 2000 SIGNALS AND SYSTEMS

• Examples of signals and systems (Electrical Systems)

– Voltage divider

• Input signal: x = 5V

• Output signal: y = Vout

• The system output is a fraction of the input (𝑦 = 𝑅2

𝑅1+𝑅2𝑥)– Multimeter

• Input: the voltage across the battery

• Output: the voltage reading on the LCD display

• The system measures the voltage across two points– Radio or cell phone

• Input: electromagnetic signals

• Output: audio signals

• The system receives electromagnetic signals and convert them to audio signal

Voltage divider

multimeter

• Examples of signals and systems (Biomedical Systems)

– Central nervous system (CNS)

• Input signal: a nerve at the finger tip senses the hightemperature, and sends a neural signal to the CNS

• Output signal: the CNS generates several output signals

to various muscles in the hand

• The system processes input neural signals, and generateoutput neural signals based on the input

– Retina

• Input signal: light

• Output signal: neural signals

• Photosensitive cells called rods and cones in the retina convert

incident light energy into signals that are carried to the brain by the optic nerve

Retina

• Examples of signals and systems (Biomedical Instrument)

– EEG (Electroencephalography) Sensors

• Input: brain signals

• Output: electrical signals

• Converts brain signal into electrical signals

– Magnetic Resonance Imaging (MRI)

• Input: when apply an oscillating magnetic field at a certain frequency, the hydrogen atoms in the body will emit radio frequency signal,

which will be captured by the MRI machine

• Output: images of a certain part of the body

• Use strong magnetic fields and radio waves to form images of the body

MRI EEG signal collection

• Signals and Systems

– Even though the various signals and systems could be quite different, they share some common properties

– In this course, we will study:

• How to represent signal and system?

• What are the properties of signals?

• What are the properties of systems?

• How to process signals with system?– The theories can be applied to any general signals and systems, be it electrical,

biomedical, mechanical, or economical, etc

SIGNALS AND CLASSIFICATIONS

• What is signal?

– Physical quantities that carry information and changes with respect to time

– E.g voice, television picture, telegraph

• Electrical signal

– Carry information with electrical parameters (e.g voltage, current)

– All signals can be converted to electrical signals

• Speech → Microphone → Electrical Signal → Speaker → Speech

– Signals changes with respect to time

audio signal

SIGNALS AND CLASSIFICATIONS

• Mathematical representation of signal:

– Signals can be represented as a function of time t

– Support of signal:

– E.g

– E.g

• and are two different signals!

– The mathematical representation of signal contains two components:

) (

) (

) (

) (

SIGNALS AND CLASSIFICATIONS

• Classification of signals: signals can be classified as

– Continuous-time signal v.s discrete-time signal

– Analog signal v.s digital signal

– Finite support v.s infinite support

– Even signal v.s odd signal

– Periodic signal v.s Aperiodic signal

– Power signal v.s Energy signal

– ……

SIGNALS: CONTINUOUS-TIME V.S DISCRETE-TIME

• Continuous-time signal

– If the signal is defined over continuous-time, then the signal is a

continuous-time signal

• E.g sinusoidal signal

• E.g voice signal

• E.g Rectangular pulse function

) 4 sin(

0

10

,)

p t

Rectangular pulse function

• Discrete-time signal

– If the time t can only take discrete values, such as,

skT

t = k = 0 ,  1 ,  2 ,  then the signal is a discrete-time signal

– E.g the monthly average precipitation at Fayetteville, AR (weather.com)

) (

) ( t s kTs

s =

– What is the value of s(t) at ?

• Discrete-time signals are undefined at !!!

month 1

=

k

SIGNALS: CONTINUOUS-TIME V.S DISCRETE-TIME

Monthly average precipitation

• Analog v.s digital

– Continuous-time signal

• continuous-time, continuous amplitude→ analog signal

– Example: speech signal

• Continuous-time, discrete amplitude

– Example: traffic light

– Discrete-time signal

• Discrete-time, discrete-amplitude → digital signal

– Example: Telegraph, text, roll a dice

0

2 1

1 0

2 3

0

2 1

Different types of signals

x ( ) =

) 2 cos(

)

) 2 sin(

)

0 ),

2 cos(

)

x

)()

(t x t

)()

x − = −

) ( )

( )

( t y t y t

y = e + o

SIGNALS: EVEN V.S ODD

• Example

– Find the even and odd decomposition of the following signal

t

e t

x( ) =

• Example

– Find the even and odd decomposition of the following signal

SIGNALS: EVEN V.S ODD

0),

4sin(

2)

x

• Periodic signal v.s aperiodic signal

– An analog signal is periodic if

• There is a positive real value T such that

• It is defined for all possible values of t, (why?)– Fundamental period : the smallest positive integer that satisfies

) ( t s t nT

) ( t = A 0t + 

SIGNALS: PERIODIC V.S APERIODIC

• Complex exponential signal

– Euler formula:

– Complex exponential signal

) sin(

) cos( x j x

ejx = +

) sin(

• The sum of two periodic signals is periodic if and only if the ratio of

the two periods can be expressed as a rational number.

• The period of the sum signal is

SIGNALS: PERIODIC V.S APERIODIC

• The sum of two periodic signals

2

T

)(

)(

)(t T ax t T by t T

• In order to have x(t)=x(t+T), T must satisfy

• In order to have y(t)=y(t+T), T must satisfy

)()

()

()

()

()

• Example

)3

2exp(

)

)9

2exp(

)

z =– Find the period of

– Is periodic? If periodic, what is the period?

– Is periodic? If periodic, what is the period?

– Is periodic? If periodic, what is the period?

)(),(),(t y t z t x

)(3)(

2x t − y t

)()(t z t

• Aperiodic signal: any signal that is not periodic

)()(t z t y

SIGNALS: PERIODIC V.S APERIODIC

SINGALS: ENERGY V.S POWER

• Signal energy

– Assume x(t) represents voltage across a resistor with resistance R.

– Current (Ohm’s law): i(t) = x(t)/R

– Instantaneous power:

– Signal power: the power of signal measured at R = 1 Ohm: p(t) = x2(t)

],

t



– Signal energy at:

t t p

p( ) = 2( )/

Instantaneous power

SINGALS: ENERGY V.S POWER

• Energy of signal x(t) over

– If then x(t) is called an 0  E   , energy signal

] ,



t

– If then x(t) is called a 0  P   , power signal

• A signal can be an energy signal, or a power signal, or neither, but not both

SINGALS: ENERGY V.S POWER

• Example 1: x ( t ) = A exp( − t )

• Example 2:

dt t

x T

0

2) ( 1

) ( t = A 0t + 

x

• Example 3: x ( t ) = ( 1 + j ) ejt 0  t  10

0( y t0

Shifting to the right by two units

OPERATIONS: SHIFTING

• Example

o.w.

3 2

2 0

0 1

0 3 1

1 )

t

t t

x

– Find x ( + t 3 )

t x(t)

-3 -2 -1 1

-1

1 2

t x(-t)

Reflection

0 1

0 1

1 )

t t

t x – Find x(3-t)

• The operations are always performed w.r.t the time variable t directly!

OPERATIONS: TIME-SCALING

• Time-scaling operation

– is obtained by scaling the signal x(t) in time.

• , signal shrinks in time domain

• , signal expands in time domain

1 2

t x(t)

-1.5 -1 -0.5 0.5 1 1.5

1 2

t x(2t)

1 2

t x(t/2)

Time scaling

OPERATIONS: TIME-SCALING

• Example:

o.w.

3 2

2 0

0 1

0 3 1

1 )

−

+

=

t t t

t

t t

x

) ( at b

x + 1 scale the signal by a: y(t) = x(at)

2 left shift the signal by b/a: z(t) = y(t+b/a) = x(a(t+b/a))=x(at+b)

• The operations are always performed w.r.t the time variable t directly (be

careful about –t or at)!

) 6 3

( t −

x

– Find

ELEMENTARY SIGNALS: UNIT STEP FUNCTION

• Unit step function

0

0 ,

0

,

1 )

2 2

,

1 )

p

• Example: rectangular pulse

Express as a function of u(t) p(t )

1

1

t u(t)

t u(t)

ELEMENTARY SIGNALS: RAMP FUNCTION

• The Ramp function

) ( )

ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION

• Unit impulse function (Dirac delta function)

1)

(

0,

0)(

)0(

t

t dt

t

t t

ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION

• Sampling property

−+ x ( t )  ( t − t0) dt = x ( t0)

) (

) ( )

( ) ( t t t0 x t0 t t0

• Shifting property

– Proof:

ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION

( – Proof:

ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION

• Examples

=

− +

ELEMENTARY SIGNALS: SAMPLING FUNCTION

• Sampling function

x

x x

Sa ( ) = sin

– Sampling function can be viewed as scaled version of sinc(x)

) (

sin )

1

t sinc(t)

Sampling function

Sinc function

ELEMENTARY SIGNALS: COMPLEX EXPONENTIAL

e t

x ( ) = ( + 0)

)]

4 (

) 2 (

[ )

t u t

u e

t